How many degrees is the altitude in a triangle? Basic elements of triangle abc. Collection and use of personal information

It is almost never possible to determine all the parameters of a triangle without additional constructions. These constructions are unique graphic characteristics of a triangle, which help determine the size of the sides and angles.

Definition

One of these characteristics is the height of the triangle. Altitude is a perpendicular drawn from the vertex of a triangle to its opposite side. A vertex is one of the three points that, together with the three sides, make up a triangle.

The definition of the height of a triangle may sound like this: the height is the perpendicular drawn from the vertex of the triangle to the straight line containing the opposite side.

This definition sounds more complicated, but it more accurately reflects the situation. The point is that in obtuse triangle It will not be possible to draw the height inside the triangle. As can be seen in Figure 1, the height in this case is external. In addition, it is not a standard situation to construct the height in a right triangle. In this case, two of the three altitudes of the triangle will pass through the legs, and the third from the vertex to the hypotenuse.

Rice. 1. Height of an obtuse triangle.

Typically, the height of a triangle is designated by the letter h. Height is also indicated in other figures.

How to find the height of a triangle?

There are three standard ways to find the height of a triangle:

Through the Pythagorean theorem

This method is used for equilateral and isosceles triangles. Let's look at the solution for isosceles triangle, and then we’ll say why the same solution is valid for an equilateral one.

Given: isosceles triangle ABC with base AC. AB=5, AC=8. Find the height of the triangle.

Rice. 2. Drawing for the problem.

For an isosceles triangle, it is important to know which side is the base. This determines sides, which must be equal, as well as the height at which some properties act.

Properties of the altitude of an isosceles triangle drawn to the base:

• The height coincides with the median and bisector
• Divides the base into two equal parts.

We denote the height as ВD. We find DC as half of the base, since the height of point D divides the base in half. DC=4

The height is a perpendicular, which means BDC is a right triangle, and the height BH is a leg of this triangle.

Let's find the height using the Pythagorean theorem: $$ВD=\sqrt(BC^2-HC^2)=\sqrt(25-16)=3$$

Any equilateral triangle is isosceles, only its base is equal to its sides. That is, you can use the same procedure.

Through the area of ​​a triangle

This method can be used for any triangle. To use it, you need to know the area of ​​the triangle and the side to which the height is drawn.

The heights in a triangle are not equal, so for the corresponding side it will be possible to calculate the corresponding height.

Triangle area formula: $$S=(1\over2)*bh$$, where b is side of the triangle, a h is the height drawn to this side. Let's express the height from the formula:

$$h=2*(S\over b)$$

If the area is 15, the side is 5, then the height is $$h=2*(15\over5)=6$$

Through the trigonometric function

The third method is suitable if the side and angle at the base are known. To do this you will have to use the trigonometric function.

Rice. 3. Drawing for the problem.

Angle ВСН=300, and side BC=8. We still have the same right triangle BCH. Let's use sine. Sine is the ratio of the opposite side to the hypotenuse, which means: BH/BC=cos BCH.

The angle is known, as is the side. Let's express the height of the triangle:

$$BH=BC*\cos (60\unicode(xb0))=8*(1\over2)=4$$

The cosine value is generally taken from the Bradis tables, but the values trigonometric functions for 30.45 and 60 degrees - tabular numbers.

What have we learned?

We learned what the height of a triangle is, what heights there are and how they are designated. Figured it out typical tasks and wrote down three formulas for the height of a triangle.

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Triangle) or pass outside the triangle at an obtuse triangle.

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Subtitles

Properties of the point of intersection of three altitudes of a triangle (orthocenter)

E A → ⋅ B C → + E B → ⋅ C A → + E C → ⋅ A B → = 0 (\displaystyle (\overrightarrow (EA))\cdot (\overrightarrow (BC))+(\overrightarrow (EB))\cdot (\ overrightarrow (CA))+(\overrightarrow (EC))\cdot (\overrightarrow (AB))=0)

(To prove the identity, you should use the formulas

A B → = E B → − E A → , B C → = E C → − E B → , C A → = E A → − E C → (\displaystyle (\overrightarrow (AB))=(\overrightarrow (EB))-(\overrightarrow (EA )),\,(\overrightarrow (BC))=(\overrightarrow (EC))-(\overrightarrow (EB)),\,(\overrightarrow (CA))=(\overrightarrow (EA))-(\overrightarrow (EC)))

Point E should be taken as the intersection of two altitudes of the triangle.)

• Orthocenter isogonally conjugate to the center circumscribed circle .
• Orthocenter lies on the same line as the centroid, the center circumcircle and the center of a circle of nine points (see Euler’s straight line).
• Orthocenter of an acute triangle is the center of the circle inscribed in its orthotriangle.
• The center of a triangle described by the orthocenter with vertices at the midpoints of the sides of the given triangle. The last triangle is called the complementary triangle to the first triangle.
• The last property can be formulated as follows: The center of the circle circumscribed about the triangle serves orthocenter additional triangle.
• Points, symmetrical orthocenter of a triangle with respect to its sides lie on the circumcircle.
• Points, symmetrical orthocenter triangles relative to the midpoints of the sides also lie on the circumscribed circle and coincide with points diametrically opposite to the corresponding vertices.
• If O is the center of the circumcircle ΔABC, then O H → = O A → + O B → + O C → (\displaystyle (\overrightarrow (OH))=(\overrightarrow (OA))+(\overrightarrow (OB))+(\overrightarrow (OC))) ,
• The distance from the vertex of the triangle to the orthocenter is twice as great as the distance from the center of the circumcircle to the opposite side.
• Any segment drawn from orthocenter Before intersecting with the circumcircle, it is always divided in half by the Euler circle. Orthocenter is the homothety center of these two circles.
• Hamilton's theorem. Three straight line segments connecting the orthocenter with the vertices of an acute triangle split it into three triangles having the same Euler circle (circle of nine points) as the original acute triangle.
• Corollaries of Hamilton's theorem:
  • Three straight line segments connecting the orthocenter with the vertices of an acute triangle divide it into three Hamilton triangle having equal radii of circumscribed circles.
  • The radii of circumscribed circles of three Hamilton triangles equal to the radius of the circle circumscribed about the original acute triangle.
• IN acute triangle the orthocenter lies inside the triangle; in an obtuse angle - outside the triangle; in a rectangular - at the top right angle.

Properties of altitudes of an isosceles triangle

• If two altitudes in a triangle are equal, then the triangle is isosceles (the Steiner-Lemus theorem), and the third altitude is both the median and the bisector of the angle from which it emerges.
• The converse is also true: in an isosceles triangle, two altitudes are equal, and the third altitude is both the median and the bisector.
• An equilateral triangle has all three heights equal.

Properties of the bases of altitudes of a triangle

• Grounds heights form a so-called orthotriangle, which has its own properties.
• The circle circumscribed about an orthotriangle is the Euler circle. This circle also contains three midpoints of the sides of the triangle and three midpoints of three segments connecting the orthocenter with the vertices of the triangle.
• Another formulation of the last property:
  • Euler's theorem for a circle of nine points. Grounds three heights arbitrary triangle, the midpoints of its three sides ( the foundations of its internal medians) and the midpoints of three segments connecting its vertices with the orthocenter, all lie on the same circle (on nine point circle).
• Theorem. In any triangle, the segment connecting grounds two heights triangle, cuts off a triangle similar to the given one.
• Theorem. In a triangle, the segment connecting grounds two heights triangles lying on two sides antiparallel to a third party with whom he has no common ground. A circle can always be drawn through its two ends, as well as through the two vertices of the third mentioned side.

Other properties of triangle altitudes

• If a triangle versatile (scalene), then it internal the bisector drawn from any vertex lies between internal median and height drawn from the same vertex.
• The height of a triangle is isogonally conjugate to the diameter (radius) circumscribed circle, drawn from the same vertex.
• In an acute triangle there are two heights cut off similar triangles from it.
• In a right triangle height, drawn from the vertex of a right angle, splits it into two triangles similar to the original one.

Properties of the minimum altitude of a triangle

The minimum altitude of a triangle has many extreme properties. For example:

• The minimum orthogonal projection of a triangle onto lines lying in the plane of the triangle has a length equal to the smallest of its altitudes.
• The minimum straight cut in a plane through which a rigid triangular plate can be pulled must have a length equal to the smallest of the heights of this plate.
• With the continuous movement of two points along the perimeter of the triangle towards each other, the maximum distance between them during the movement from the first meeting to the second cannot be less than the length of the smallest height of the triangle.
• The minimum height in a triangle always lies within that triangle.

Basic relationships

• h a = b ⋅ sin ⁡ γ = c ⋅ sin ⁡ β , (\displaystyle h_(a)=b(\cdot )\sin \gamma =c(\cdot )\sin \beta ,)
• h a = 2 ⋅ S a , (\displaystyle h_(a)=(\frac (2(\cdot )S)(a)),) Where S (\displaystyle S)- area of ​​a triangle, a (\displaystyle a)- the length of the side of the triangle by which the height is lowered.
• h a = b ⋅ c 2 ⋅ R , (\displaystyle h_(a)=(\frac (b(\cdot )c)(2(\cdot )R)),) Where b ⋅ c (\displaystyle b(\cdot )c)- product of the sides, R − (\displaystyle R-) circumscribed circle radius
• h a: h b: h c = 1 a: 1 b: 1 c = (b ⋅ c) : (a ⋅ c) : (a ⋅ b) .
• (\displaystyle h_(a):h_(b):h_(c)=(\frac (1)(a)):(\frac (1)(b)):(\frac (1)(c)) =(b(\cdot )c):(a(\cdot )c):(a(\cdot )b).) 1 h a + 1 h b + 1 h c = 1 r (\displaystyle (\frac (1)(h_(a)))+(\frac (1)(h_(b)))+(\frac (1)(h_ (c)))=(\frac (1)(r))) , Where r (\displaystyle r)
• - radius of the inscribed circle. 1 h a + 1 h b + 1 h c = 1 r (\displaystyle (\frac (1)(h_(a)))+(\frac (1)(h_(b)))+(\frac (1)(h_ (c)))=(\frac (1)(r))) S (\displaystyle S) - S = 1 (1 h a + 1 h b + 1 h c) ⋅ (1 h a + 1 h b − 1 h c) ⋅ (1 h a + 1 h c − 1 h b) ⋅ (1 h b + 1 h c − 1 h a) (\displaystyle S =(\frac (1)(\sqrt (((\frac (1)(h_(a)))+(\frac (1)(h_(b)))+(\frac (1)(h_(c ))))(\cdot )((\frac (1)(h_(a)))+(\frac (1)(h_(b)))-(\frac (1)(h_(c))) )(\cdot )((\frac (1)(h_(a)))+(\frac (1)(h_(c)))-(\frac (1)(h_(b))))(\ cdot )((\frac (1)(h_(b)))+(\frac (1)(h_(c)))-(\frac (1)(h_(a)))))))).
• a = 2 h a ⋅ (1 h a + 1 h b + 1 h c) ⋅ (1 h a + 1 h b − 1 h c) ⋅ (1 h a + 1 h c − 1 h b) ⋅ (1 h b + 1 h c − 1 h a) (\ displaystyle a=(\frac (2)(h_(a)(\cdot )(\sqrt (((\frac (1)(h_(a)))+(\frac (1)(h_(b))) +(\frac (1)(h_(c))))(\cdot )((\frac (1)(h_(a)))+(\frac (1)(h_(b)))-(\ frac (1)(h_(c))))(\cdot )((\frac (1)(h_(a)))+(\frac (1)(h_(c)))-(\frac (1 )(h_(b))))(\cdot )((\frac (1)(h_(b)))+(\frac (1)(h_(c)))-(\frac (1)(h_ (a))))))))), a (\displaystyle a)- the side of the triangle to which the height descends h a (\displaystyle h_(a)).
• Height of an isosceles triangle lowered to the base: h c = 1 2 ⋅ 4 a 2 − c 2 , (\displaystyle h_(c)=(\frac (1)(2))(\cdot )(\sqrt (4a^(2)-c^(2)) ),)

Where c (\displaystyle c)- base, a (\displaystyle a)- side.

Right Triangle Altitude Theorem

If the altitude in a right triangle ABC is of length h (\displaystyle h) drawn from the vertex of a right angle, divides the hypotenuse with length c (\displaystyle c) into segments m (\displaystyle m) And n (\displaystyle n), corresponding to the legs b (\displaystyle b) And a (\displaystyle a), then the following equalities are true.

To solve many geometric problems you need to find the height of a given figure. These tasks have practical significance. When carrying out construction work, determining the height helps to calculate the required amount of materials, as well as determine how accurately slopes and openings are made. Often, to create patterns, you need to have an idea of ​​the properties

Many people, despite good grades at school, when constructing ordinary geometric figures, have a question about how to find the height of a triangle or parallelogram. And it is the most difficult. This is because a triangle can be acute, obtuse, isosceles or right. Each of them has its own rules of construction and calculation.

How to find the height of a triangle in which all angles are acute, graphically

If all the angles of a triangle are acute (each angle in the triangle is less than 90 degrees), then to find the height you need to do the following.

1. Using the given parameters, we construct a triangle.
2. Let us introduce some notation. A, B and C will be the vertices of the figure. The angles corresponding to each vertex are α, β, γ. The sides opposite these angles are a, b, c.
3. The altitude is the perpendicular drawn from the vertex of the angle to the opposite side of the triangle. To find the heights of a triangle, we construct perpendiculars: from the vertex of angle α to side a, from the vertex of angle β to side b, and so on.
4. Let us denote the intersection point of the height and side a as H1, and the height itself as h1. The intersection point of the height and side b will be H2, the height, respectively, h2. For side c, the height will be h3 and the intersection point will be H3.

Height in a triangle with an obtuse angle

Now let's look at how to find the height of a triangle if there is one (more than 90 degrees). In this case, the height drawn from obtuse angle, will be inside the triangle. The remaining two heights will be outside the triangle.

Let the angles α and β in our triangle be acute, and the angle γ be obtuse. Then, to construct the heights coming from the angles α and β, it is necessary to continue the sides of the triangle opposite them in order to draw perpendiculars.

How to find the height of an isosceles triangle

This figure has two equal sides and the base, while the angles located at the base are also equal to each other. This equality of sides and angles makes it easier to construct heights and calculate them.

First, let's draw the triangle itself. Let the sides b and c, as well as the angles β, γ, be equal, respectively.

Now let’s draw the height from the vertex of angle α, denoting it h1. For this height will be both a bisector and a median.

Only one construction can be made for the foundation. For example, draw a median - a segment connecting the vertex of an isosceles triangle and the opposite side, the base, to find the height and bisector. And to calculate the length of the height for the other two sides, you can construct only one height. Thus, to graphically determine how to calculate the height of an isosceles triangle, it is enough to find two of the three heights.

How to find the height of a right triangle

For a right triangle, determining the heights is much easier than for others. This happens because the legs themselves make a right angle, and therefore are heights.

To construct the third height, as usual, a perpendicular is drawn connecting the vertex of the right angle and the opposite side. As a result, in order to create a triangle in this case, only one construction is required.

When solving geometric problems, it is useful to follow such an algorithm. While reading the conditions of the problem, it is necessary

• Make a drawing. The drawing should correspond as much as possible to the conditions of the problem, so its main task is to help find the solution
• Put all the data from the problem statement on the drawing
• Write down all the geometric concepts that appear in the problem
• Remember all the theorems that relate to these concepts
• Draw on the drawing all the relationships between elements geometric figure, which follow from these theorems

For example, if the problem contains the words bisector of an angle of a triangle, you need to remember the definition and properties of a bisector and indicate equal or proportional segments and angles in the drawing.

In this article you will find the basic properties of a triangle that you need to know to successfully solve problems.

TRIANGLE.

Area of ​​a triangle.

1. ,

here - an arbitrary side of the triangle, - the height lowered to this side.

2. ,

here and are arbitrary sides of the triangle, and is the angle between these sides:

3. Heron's formula:

Here are the lengths of the sides of the triangle, is the semi-perimeter of the triangle,

4. ,

here is the semi-perimeter of the triangle, and is the radius of the inscribed circle.

Let be the lengths of the tangent segments.

Then Heron's formula can be written as follows:

5.

6. ,

here - the lengths of the sides of the triangle, - the radius of the circumscribed circle.

If a point is taken on the side of a triangle that divides this side in the ratio m: n, then the segment connecting this point with the vertex of the opposite angle divides the triangle into two triangles, the areas of which are in the ratio m: n:

The ratio of the areas of similar triangles is equal to the square of the similarity coefficient.

Median of a triangle

This is a segment connecting the vertex of a triangle to the middle of the opposite side.

Medians of a triangle intersect at one point and are divided by the intersection point in a ratio of 2:1, counting from the vertex.

The intersection point of the medians of a regular triangle divides the median into two segments, the smaller of which is equal to the radius of the inscribed circle, and the larger of which is equal to the radius of the circumscribed circle.

The radius of the circumscribed circle is twice the radius of the inscribed circle: R=2r

Median length arbitrary triangle

,

here - the median drawn to the side - the lengths of the sides of the triangle.

Bisector of a triangle

This is the bisector segment of any angle of a triangle connecting the vertex of this angle with the opposite side.

Bisector of a triangle divides a side into segments proportional to the adjacent sides:

Bisectors of a triangle intersect at one point, which is the center of the inscribed circle.

All points of the angle bisector are equidistant from the sides of the angle.

Triangle height

This is a perpendicular segment dropped from the vertex of the triangle to the opposite side, or its continuation. In an obtuse triangle, the altitude drawn from the vertex of the acute angle lies outside the triangle.

The altitudes of a triangle intersect at one point, which is called orthocenter of the triangle.

To find the height of a triangle drawn to the side, you need to find its area in any available way, and then use the formula:

Center of the circumcircle of a triangle, lies at the intersection point perpendicular bisectors drawn to the sides of the triangle.

Circumference radius of a triangle can be found using the following formulas:

Here are the lengths of the sides of the triangle, and is the area of ​​the triangle.

,

where is the length of the side of the triangle and is the opposite angle. (This formula follows from the sine theorem.)

Triangle inequality

Each side of the triangle is less than the sum and greater than the difference of the other two.

The sum of the lengths of any two sides is always longer third party:

Opposite the larger side lies the larger angle; Opposite the larger angle lies the larger side:

If , then vice versa.

Theorem of sines:

The sides of a triangle are proportional to the sines of the opposite angles:

Cosine theorem:

The square of a side of a triangle is equal to the sum of the squares of the other two sides without twice the product of these sides by the cosine of the angle between them:

Right triangle

- This is a triangle, one of the angles of which is 90°.

The sum of the acute angles of a right triangle is 90°.

The hypotenuse is the side that lies opposite the 90° angle. The hypotenuse is the longest side.

Pythagorean theorem:

the square of the hypotenuse is equal to the sum of the squares of the legs:

The radius of a circle inscribed in a right triangle is equal to

,

here is the radius of the inscribed circle, - the legs, - the hypotenuse:

Center of the circumcircle of a right triangle lies in the middle of the hypotenuse:

Median of a right triangle drawn to the hypotenuse, is equal to half the hypotenuse.

Definition of sine, cosine, tangent and cotangent of a right triangle look

The ratio of elements in a right triangle:

The square of the altitude of a right triangle drawn from the vertex of a right angle is equal to the product of the projections of the legs onto the hypotenuse:

The square of the leg is equal to the product of the hypotenuse and the projection of the leg onto the hypotenuse:

Leg lying opposite the corner equal to half the hypotenuse:

Isosceles triangle.

The bisector of an isosceles triangle drawn to the base is the median and altitude.

In an isosceles triangle, the base angles are equal.

Apex angle.

And - sides,

And - angles at the base.

Height, bisector and median.

Attention! The height, bisector and median drawn to the side do not coincide.

Regular triangle

(or equilateral triangle ) is a triangle, all sides and angles of which are equal to each other.

Area of ​​a regular triangle equal to

where is the length of the side of the triangle.

Center of a circle inscribed in a regular triangle, coincides with the center of the circle circumscribed about a regular triangle and lies at the point of intersection of the medians.

Intersection point of the medians of a regular triangle divides the median into two segments, the smaller of which is equal to the radius of the inscribed circle, and the larger of which is equal to the radius of the circumscribed circle.

If one of the angles of an isosceles triangle is 60°, then the triangle is regular.

Middle line of the triangle

This is a segment connecting the midpoints of two sides.

In the figure DE is the middle line of triangle ABC.

The middle line of the triangle is parallel to the third side and equal to its half: DE||AC, AC=2DE

External angle of a triangle

This is the angle adjacent to any angle of the triangle.

An exterior angle of a triangle is equal to the sum of two angles not adjacent to it.

External angle trigonometric functions:

Signs of equality of triangles:

1 . If two sides and the angle between them of one triangle are respectively equal to two sides and the angle between them of another triangle, then such triangles are congruent.

2 . If a side and two adjacent angles of one triangle are respectively equal to a side and two adjacent angles of another triangle, then such triangles are congruent.

3 If three sides of one triangle are respectively equal to three sides of another triangle, then such triangles are congruent.

Important: since in a right triangle two angles are obviously equal, then for equality of two right triangles equality of only two elements is required: two sides, or a side and an acute angle.

Signs of similarity of triangles:

1 . If two sides of one triangle are proportional to two sides of another triangle, and the angles between these sides are equal, then these triangles are similar.

2 . If three sides of one triangle are proportional to three sides of another triangle, then the triangles are similar.

3 . If two angles of one triangle are equal to two angles of another triangle, then the triangles are similar.

Important: V similar triangles similar sides lie opposite equal angles.

Menelaus' theorem

Let a line intersect a triangle, and is the point of its intersection with side , is the point of its intersection with side , and is the point of its intersection with the continuation of side . Then